The study of discrete nodal domain theorems on graphs dates back to the work of Gantmacher and Krein [1]}, which contains a discrete analogue of Sturm's theorem for strings. Many of Fiedler's results in 1970s [2]}, [3]}, [4]} can be interpreted as discrete nodal domain estimates. The discrete nodal domain theorems for generalized Laplacians were eventually established by Davies, Gladwell, Leydold, and Stadler [5]} in 2001. There are many further advances in this topic, see, e.g., [6]}, [7]}, [8]} and the book [9]}.
